It's correct that the 70-300mm f/4-5.6 lenses cannot use Canon teleconverters (neither the non-L nor the L version). 3rd party teleconverters can be used, but as an f/4-5.6 lens, you'll have problems with autofocus if you use it with a TC.
But...what is your budget? If by 'cheapish' you mean under $1K, your selection of Canon lenses that can use a TC is limited to one zoom lens, the 70-200mm f/4L, and two prime lenses, the 200mm f/2.8L II and possibly the 135mm f/2L (just over $1K with the current rebate).
If you are looking at the 70-300mm non-L, I'd recommend the 70-200mm f/4L instead. With a Canon 1.4x II, that will still come in under $1K, give you a 70-200/4 or a 98-280/5.6, and better IQ than the 70-300mm non-L. But, if your budget is higher, Amazon has the 70-300mm f/4-5.6L IS for under $1400, and will give better IQ and more convenience than the 70-200/4+1.4x (and though the 70-300 L doesn't take TCs, it still gives a broader range at approximately equivalent apertures).